Weighted composition operators on the logarithmic Bloch-Orlicz space

The boundedness and compactness of weighted composition operators on the logarithmic Bloch-Orlicz space Blogφ are investigated in this paper.


Introduction
Suppose that S(D) is the set of all analytic self-maps of the unit disk D (the analytic maps from D to itself).Let H(D) be, as usual, the collection of all analytic functions on D. The composition operator, induced by an analytic self-map of the unit disk ϕ, could be defined as For ψ ∈ H(D), the weighted composition operator could be defined as The study of composition operators and weighted composition operators has been established for over 6 decades.Many mathematicians aimed at the research of (weighted) composition operators on spaces of analytic functions on D or on some high-dimension domains (e.g., the unit ball of C n , the unit polydisk of C n ).We refer to reference [3] for studying the history of composition operators acting on different spaces of analytic functions.
As one of the classical space of analytic functions on D, the Bloch space, is defined as Importantly, B is maximal among all Möbius-invariant spaces of analytic functions and B turns to a complete normed linear space endowed with the norm The logarithmic Bloch space B log is defined as It is a Banach space endowed with the norm f log = |f (0)| + f B log .The µ−Bloch space B µ is defined as where the weight function µ(z) is a positive continuous function in D. Then B µ is a Banach space endowed with the norm In the recent years, the boundedness and compactness of composition operators among different Bloch(-type) spaces were studied (see, e.g., [5,15,16] and the references therein).Moreover, as the study of Hardy-Orlicz space and Bergman-Orlicz space (see, e.g., [2,6,[9][10][11][12]14]), the Bloch-Orlicz space B ϕ was defined as a generalization of B.
It was firstly defined in [4], for some λ > 0 depending of f , where ϕ is the Young's function.Basic properties of the convex function imply We can assume without loss of generality that ϕ −1 is differentiable (as the authors did in [4]).Suppose that defines a semi-norm for B ϕ .In this way, B ϕ is a Banach space endowed with the norm In this paper, we generalize the Bloch-Orlicz space to the logarithmic Bloch-Orlicz space.Note that it is a generalization of the logarithmic Bloch space B log .The logarithmic Bloch-Orlicz space B ϕ log is defined as follows, for some λ > 0 depending of f , where ϕ is the Young's function and ϕ −1 is differentiable.Moreover, defines a semi-norm for B ϕ log , where Thus, B ϕ log becomes a Banach space endowed with the norm . This paper is organized as follows: in Section 2 we recall some basic facts on the logarithmic Bloch-Orlicz space.In Section 3 we investigate the boundedness of weighted composition operators on the logarithmic Bloch-Orlicz space.Moreover, in Section 4 we investigate the compactness of weighted composition operators on the logarithmic Bloch-Orlicz space.
We say that ϕ ∈ U if there exists a constant M ϕ > 1 such that which also implies that

Preliminaries
In this section, we present several basic conclusions for the study of B ϕ log .
Proof.The proof is similar with Lemma 2 in [4].
For each f ∈ B ϕ log \ {0}, a decreasing sequence {λ n,log } n of positive numbers can be chosen, satisfying lim n→∞ λ n,log = f ϕ,log and S ϕ,log ( f λ n,log ) ≤ 1.For any positive integer n ∈ N, let S n := S ϕ,log ( f λ n,log ).Observe that {S n } is increasing and bounded and hence there exists a real number S ∈ R satisfying lim n→∞ S n = S .It follows that Thus we have S ≤ S.Moreover, ) ≤ S ≤ 1 holds for each z ∈ D and n ∈ N. Taking limit as n → ∞, the above inequality becomes for each z ∈ D, which is equivalent to say that This completes the proof.
holds for all f ∈ B ϕ log and z ∈ D by Lemma 2.1, where M ϕ > 1 is a constant only dependent of ϕ.In fact, a simple estimation shows that where more details can be found in [17].The inequality above also implies that the evaluation functional is continuous on B ϕ log , where z ∈ D is fixed.
The proposition below shows that the logarithmic Bloch-Orlicz space is isometrically equal to a µ−Bloch space.

Proposition 2.3
The logarithmic Bloch-Orlicz space is isometrically equal to a µ log −Bloch space, where In other words, Proof.Deducted by Proposition 2.1, for each f ∈ B ϕ log and z ∈ D, Proof.The sufficiency part is obvious.The necessity is deducted by Proposition 2.1 and the estimation S ϕ,log (f ) ≤ S ϕ,log ( f f ϕ,log ) ≤ 1.For two real numbers A and B, we say A B if there exists a constant C = 0 such that A ≤ CB, by which the complexity of all constants appearing is simplified.

Boundedness of Weighted Composition Operator on B ϕ log
In this section we investigate the boundedness of weighted composition operators on the logarithmic Bloch-Orlicz space under the condition ϕ ∈ U(this is an unexpected hypothesis).However, for those Young's funtion ϕ / ∈ U, the boundedness of weighted composition operators on the logarithmic Bloch-Orlicz space remains to be an open question.
The first lemma contains some trivial but complicated calculations, which will be used in the proof in this section.
Basic properties of the auxiliary functions entailed to the proof of the boundedness of weighted composition operator ψC φ are described in the next lemma.and where z ∈ D. Then the auxiliary functions p a and q a have the properties as follows: (i) the auxiliary function p a belongs to the logarithmic Bloch-Orlicz space B ϕ log with sup a∈D p a B ϕ log 1.
(ii) the auxiliary function q a belongs to the logarithmic Bloch-Orlicz space B ϕ log with sup a∈D q a B ϕ log 1.
Proof.As easy calculation shows, Observe that Then we conclude that sup a∈D S ϕ,log (q a ) 1. Further observe that, by Lemma 3.1 Then we conclude that sup a∈D S ϕ,log (p a ) 2.
Though the approach we use in the proof of the boundedness is standard, it is not trivial since the collection U contains not only (almost) linear functions, it also contains the convex functions which line between the the line whose tangent is 1 and a line whose tangent is M ϕ , a positive number depending on ϕ.Theorem 3.3 For ϕ ∈ U, the weighted composition operator ψC φ is bounded on B ϕ log if and only if February 4, 2024 6/12 Taking f 0 (z) = 1 ∈ B ϕ log , employing (3.1) we obtain that Further taking f0 = z ∈ B ϕ log , employing (3.1) again we obtain that and hence by L 1 < ∞ we conclude that On the other hand, for each a, z ∈ D, we define For each a ∈ D, we have Combining what we have observed above, we complete the proof.

Compactness of Weighted Composition Operator on B ϕ log
In this section we investigate the compactness of weighted composition operators on the logarithmic Bloch-Orlicz space, where the approach we use in the proof within is standard (see, e.g., [7]).The first lemma contains some trivial but complicated calculations, which will be used in the proof of the compactness of weighted composition operator on B ϕ log .Moreover, it can be proved in a similar way with Lemma 3. Theorem 4.2 For ϕ ∈ U, the weighted composition operator ψC φ is compact on B ϕ log if and only if ψC φ is bounded on B ϕ log , and Proof.Suppose that the weighted composition operator ψC φ is bounded on B ϕ log and (4.1) (4.2) hold.Note that L 1 < ∞ and L 2 < ∞ defined in the proof of Theorem 3.3 (see, (3.2) and (3.3), respectively) by the boundedness of ψC φ .For every > 0, there exists an 0 < r < 1 such that for |φ(z)| > r, hold.For a chosen sequence {f n } n ⊂ B ϕ log that satisfy sup n∈N f n B ϕ log ≤ K and {f n } converges to zero uniformly on any compact subsets of the unit disk as n → ∞, where K is a fixed constant.It is sufficient to show that lim n→∞ ψC φ f n B ϕ log = 0 by the compactness of ψC φ .Note that lim n→∞ f n (0) = 0 and {f n } converges to zero uniformly on any compact subsets of the unit disk.It follows by Proposition 2.3 that Then we conclude that ψC φ is compact on B ϕ log by the arbitrariness of > 0. Conversely, suppose that ψC φ is compact on B ϕ log and hence ψC φ is bounded on B ϕ log .We prove (4.1) and (4.2) hold as follows.Let {z n,log } n be a sequence in the unit disk satisfying lim n→∞ |φ(z n,log )| = 1.If such sequence does not exist, then the proof is completed.
On the one hand, for each z ∈ D, we consider the function pφ(z n,log ) (z), where pa is constructed in Lemma 4.1.Then we have p φ(z n,log ) (φ(z n,log )) = 0 and Note that {p φ(z n,log ) } n is bounded uniformly in B ϕ log and uniformly converges to zero on any compact subset of the unit disk as n → ∞ by (1.1) .Thus we have lim On the other hand, we consider the function q φ(z n,log ) (z), where q a is constructed in Lemma 3.2.Then we have q φ(z n,log ) (φ(z n,log )) = 0 and Note that {q φ(z n,log ) } n is bounded uniformly in B ϕ log and uniformly converges to zero on any compact subset of the unit disk as n → ∞ since  Combining what we have observed above, we complete the proof.

Boundedness and Compactness of Weighted Composition Operators on B µ log
Recall that it is proved in Proposition 2.3 that the logarithmic Bloch-Orlicz space is isometrically equal to a µ log −Bloch space, where Hence, the boundedness and compactness of weighted composition operators on the µ log −Bloch space B µ log could be naturally given.

Lemma 4 . 1
For a ∈ D, n ∈ N + and ϕ ∈ U, suppose that pa ((a)| 2 ) 2 where z ∈ D. Then the auxiliary function pa belongs to the logarithmic Bloch-Orlicz space B ϕ log with sup a∈D pa B log 1.
Therefore, we obtain that B µ log ⊂ B ϕ log with f ϕ,log ≤ f µ log .Combining what we have observed above, we complete the proof.
and the second inequality is deducted by (2.1).Then we conclude that ψC φ is bounded on B ϕ log byψC φ f ϕ,log ≤ C f B ϕ log and the estimation (2.1) by taking z = φ(0).Conversely, if ψC φ is bounded on B ϕ log , then there exists a constant C ≥ 0 such thatψC φ f B ϕ log ≤ C f B ϕ log for each 0 = f ∈ B ϕ log .By Proposition 2.1 and the condition above,